Martin Kneser (1928 – 2004)
Martin Kneser’s Work on Quadratic Forms and
Algebraic Groups
- strong approximation and class
numbers
Rudolf Scharlau
- Galois cohomology of algebraic
groups
International Conference on the Algebraic and Arithmetic Theory of
Quadratic Forms
- geometry of numbers, explicit
constructions of lattices
Llanquihue, Chile, 19.12.2007
2
1
Contents
1 A Short History of Quadratic Forms 1884 – 1954
4
1
2 Martin Kneser: Quadratic forms and arithmetic of algebraic
groups 1955 – 1970
20
2.1
Strong approximation and class numbers . . . . . . . . 28
2.2
Local-global principles and Galois cohomology . . . . . 32
2.3
Siegel’s theorem and Tamagawa numbers . . . . . . . . 34
2.4
Further contributions by M. Kneser . . . . . . . . . . . 37
A Short History of Quadratic Forms 1884 –
1954
The theory of quadratic forms emerged as a part of (elementary)
number theory, dealing with quadratic diophantine equations, initially
over the rational integers.
3 A particular problem: finite quotients of Bruhat-Tits buildings 40
3
4
Herrmann Minkowski (1864 – 1909)
The main questions in modern language were:
a. the equivalence problem: when are two quadratic modules
(“lattices”) (L, q) und (M, q ′ ) over Z equivalent ?
b. The representation problem: for which t ∈ Z does there exist a
x ∈ L with q(x) = t ?
c. The determination of the representation numbers
a(t, L) = |{x ∈ L | q(x) = t}|.
5
He developed the foundations of a
general theory of quadratic forms over
the rationals and rational integers. He
already proved major results on all
three questions in a modern way.
6
In his Königsberg Dissertation from 1885 entitled
A brilliant work of the very young Minkowski is the prize-winning
paper
Grundlagen für eine Theorie der quadratischen Formen mit ganzzahligen Koeffizienten, Mémoires présentés par divers savants a
l’Académie des Sciences de l’institut national de France, Tome
XXIX, No. 2. 1884.
In the main part of this paper, he develops the local classification of
integral quadratic forms. In the context of the prize question on sums
of five squares, this was preparatory, but clearly of independent
importance.
7
Untersuchungen über quadratische Formen. Bestimmung der
Anzahl verschiedener Formen, die ein gegebenes Genus enthält.
Königsberg 1885; Acta Mathematica 7 (1885), 201–258
he proves a version of the Maßformel which is already very similar to
the current one. In contrast to the works of previous authors, the
“right hand side” is a product of local densities over all prime
numbers.
In this context, Minkowski also introduces for the first time (more or
less) today’s notion of a genus of quadratic forms (in any number
over variables).
8
In those days, the rational theory (classification over Q) still was a
by-product of the integral theory. Nevertheless, the following paper
practically contains the main theorem over Q.
H. Minkowski (Letter to Hurwitz), Über die Bedingungen, unter
welchen zwei quadratische Formen mit rationalen Koeffizienten
ineinander rational transformiert werden können, J. reine angew.
Math. 106 (1890), 5–26 = Ges. Abh. I, 219–239.
To every rational quadratic form, Minkowski associates a system of
invariants Cp = ±1, one for each prime. He shows that these
invariants, together with the discriminant (a rational square class),
determine the rational equivalence class. This result contains the
local-global principle (for equivalence, not for representations), but
the term is not yet used.
end Minkowski
9
10
Helmut Hasse (1898 – 1979)
Carl Ludwig Siegel (1896 – 1982)
One of the leading german algebraic
number theorists in the 20th century
- analytic number theory
- introduces Hensel’s p-adic numbers into the theory of quadratic
forms
- discrete groups, complex analysis
- complete solution of the problem
of representation numbers of integral quadratic forms
- proves the (strong) local-global
principle over number fields
11
12
Theorem (Minkowski, Siegel) Es sei L ein positiv definites Gitter der
Dimension ℓ und M = M1 , . . . , Mh ein Repräsentantensystem für ein
Geschlecht positiv definiter Gitter der Dimension m. Dann gilt für die
Darstellungsanzahlen a(L, Mk ) von L durch die verschiedenen Mk und die
lokalen Darstellungsdichten αp (L, M ), p prim, die Beziehung
P
k
Ernst Witt (1911 – 1991)
- the founder of the modern
theory of quadratic forms over
arbitrary fields
X a(L, Mk )
γ(m − ℓ) Y
1
·
=
αp (L, M ).
|O(Mk )|−1
|O(Mk )|
γ(m)
- the cancellation theorem
k
- the extension theorem for
isometries
Hierbei sind die Werte γ(n) induktiv definiert durch
1
1
γ(n − 1)
γ(0) = 1, γ(1) = , γ(2) =
, γ(n) =
für m ≥ 3,
2
2π
n · ρn
Theorie der quadratischen Formen in
beliebigen Körpern, J. reine angew.
Math. 176 (1937), 31–44 = Coll. Papers, Ges. Abh. 2–15
wo ρn das Volumen der n-dimensionalen Einheitskugel ist.
Über die analytische Theorie der quadratischen Formen I, II, III, Annals of
Mathematics 36 (1935), 527–606, 37 (1936), 230–263, 38 (1937), 212–291
14
13
In particular, I want to mention Witt’s paper
Witt was a very original mathematician; he made fundamental
contributions to diverse of topics: Witt index, Witt group, Witt
vectors, to name just three. For instance, in the theory of Lie
algebras, in modular forms and in algebraic combinatorics he is cited
for some standard results.
Eine Identität zwischen Modulformen zweiten Gerades, Abh.
Math. Sem. Univ. Hamburg 14 (1941), 323–337 = Coll. Papers,
Ges. Abh. 313–328.
It is one of the first contributions to ongoing research on “lattices and
e 16 und E8 ⊥ E8 , not only the
modular forms”. He shows that for D
ordinary theta series, but also the second degree Siegel theta series
coincide.
We shall come back to this later.
end Witt
15
16
Martin Eichler (1912 – 1992)
- simple algebras over number
fields
Martin Kneser very clearly acknowledges the influence of Eichler on
his own work in the introduction of the 2001 book version of his
lectures on quadratic forms:
“Für all dies vergleiche man das einflußreiche Werk
Quadratische Formen und orthogonale Gruppen, SpringerVerlag 1952.”
- spinor norms, spinor genera
- first approximation results
- modular forms, theta series
He makes precise in what sense the book was influential on himself:
17
“Schließlich ein persönliches Wort. Es ist
ziemlich genau 50 Jahre her, daß ich als
junger Assistent nach Münster kam, bald
an Eichlers Seminar teilnahm, wo gerade
die neuesten Ergebnisse aus seinem Buch
Quadratische Formen und orthogonale Gruppen besprochen wurden. Da ich im Institut
mein Arbeitszimmer mit Eichler teilte, hatte
ich die besten Möglichkeiten, von einer Seminarsitzung zur nächsten die offen gebliebenen
Fragen zu klären und so die quadratischen
Formen an der Quelle zu studieren.”
(M. Kneser 2001, aus der Einleitung von Quadratische Formen)
19
18
2
Martin Kneser: Quadratic forms and
arithmetic of algebraic groups 1955 – 1970
In the mid 1950s, the theory of algebraic groups and the (arithmetic)
theory of quadratic forms were still rather unrelated areas of research. On
the side of groups, the classification of (semi)simple algebraic groups over
algebraically closed fields was known by work of Claude Chevalley. Jacques
Tits had (essentially) introduced the structures later called buildings which
give a uniform geometrical interpretaion of all these groups, including the
exceptional ones.
20
Already by the end of the 1950s, a completely new area of research had
emerged, after Armand Borel had proved his fundamental theorem on the
existence and conjugacy of maximal connected solvable subgroups. This
made the classification of semisimple groups over arbitrary fields
accessible, which was then rather quickly carried out mainly by Borel and
Tits. They used k-split tori and the relative root system to reduce the
question essentially to the anisotropic kernel, in analogy with the Witt
decomposition of quadratic forms.
Over number fields, this approach embedded the earlier studies of algebras
with involution, hermitian forms, Cayley octaves and Jordan algebras into
a uniform theory. In this situation it was perfectly natural (after work of
Lang and Tate) to introduce non-abelian Galois cohomology (H 0 , H 1 ,
abelian H 2 ) to treat such classification questions. Jean-Pierre Serre’s
course at the Collège de France 1962-63, leading to the famous Lecture
notes No. 5 Cohomologie Galoisienne, demonstrates how quickly the new
method had been established.
22
21
Martin Kneser’s vita
The theory of semisimple groups over number fields in turn laid the
foundations for a general treatment of arithmetic subgroups of algebraic
groups, whose fundamentals were developed by Borel and Harish-Chandra.
Clearly, many substantial results had been obtained (much) earlier mainly
by Siegel, but the framework had dramatically changed.
- 1945-50 studies in Tübingen,
Göttingen and Berlin
- 1951-1956 assistant at Münster
and Heidelberg, Habilitation
- 1957-58 Univ. of Saarbrücken
We now want to look at (part of) Kneser’s work as embedded in this
general picture.
23
- 1959-62 Prof at München
- 1963–1993 Prof at Göttingen
24
Four fundamental works by Martin Kneser:
1. Klassenzahlen indefiniter quadratischer Formen,
Archiv d. Math. 7 (1956), 323–332
2. Klassenzahlen definiter quadratischer Formen,
Archiv d. Math. 8 (1957), 241–250.
3a. Strong approximation. in: Algebraic groups and discontinuous subgroups. Proceedings, Boulder Co 1965.
3b. Starke Approximation in algebraischen Gruppen.I.
J. reine angew. Math. 218 (1965), 190 – 203.
4. Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern I. and II.
Math. Z.88 (1965), 40–47, 89 (1965), 250–272
contents of the paper in Archiv d. Math. 1956:
- proof of the strong approximation theorem for representations
and for the orthogonal group
- the adelic orthogonal group is introduced for the first time
- the number of spinor genera in a genus is a group index
- generation of orthogonal groups by reflections
- computation of local spinor norms
26
25
contents of the paper in Archiv d. Math. 1957:
- main idea: if (V, q) is isotropic at p, lattice over Z[1/p] behave
like indefinite lattices
- technically: apply strong approximation (from the previous paper)
to the set of places S = ∞ ∪ {p}
- for any two classes in the same spinor genus, there are
representatives L, M s.t. Z[1/p]L = Z[1/p]M.
- the resulting “neighbour method” is used to calculate the class
number of In up to dimension 14.
27
2.1
Strong approximation and class numbers
We have already talked, without details, about the use of strong
approximation for class numbers and representations of quadratic
forms. We now generalize the situation to algebraic groups and give
complete definitions and statements.
Notation:
28
k
an algebraic number field
o
the ring of integers of k
V
a finite-dimensional vector space over k
p, ℓ, v, . . .
places (equivalence classes of valutations of k)
G
a linear algebraic group defined over o
kp , op
the completion of k, resp. o at p
L
a lattice in V
p ∈ op
a prime element for p, if p is finite
G(R)
for any over-ring R ⊇ o the group of R-points in G
Sk
the set of all places of k
S
a finite set of places of k
G(Ak )
the adele-group of G over k
A = Ak
the ring of adeles of k
G(o)
the stabilizer of L in G(k)
(aℓ )ℓ∈Sk
a typical element of A, so aℓ ∈ ol f.a.a. ℓ
G(k) ⊂ G(Ak )
diagonally embedded
A(S) ⊂ A
the S-integral ideles, so aℓ ∈ oℓ for ℓ ∈
/S
in particular
29
30
2.2
Definition 1 The G-class number of a lattice L is the number of
G-classes in the G-genus of L.
Definition 2 (Strong Approximation) Let G be an algebraic
group over k and S be a finite set of places of k. We say that strong
approximation holds for the pair (G, S) if G(k)G(A(S)) is dense in
G(Ak ).
Theorem 1 (Kneser 1965, Platonov 1969)
Strong approximation holds for all pairs (G, S), where G is simply
connected almost k-simple and G(kv ) is not compact for at least one
v ∈ S.
31
Local-global principles and Galois cohomology
Definition 3 The Hasse principle holds for an algebraic group G over
k if the canonical map
Y
H 1 (kv , G)
H 1 (k, G) →
v∈Sk
is injective.
The following result is fundamental for the use of this method:
Theorem 2 (Kneser 1965) If G is a semisimple simply connected
group over a local field k of characteristic 0, then H 1 (k, G) = 0.
The proof uses the classification and structure theory of such groups,
but also a lot of case-by-case investigations.
32
Theorem 3 (Kneser 1965, Harder 1965/66, Chernousov 1989)
The Hasse principle holds for all semisimple simply connected
algebraic groups.
Remark: Naturally, this theorem is not restricted to simply connected
groups, as already the case of orthogonal groups shows. In particular,
it holds for all (connected) adjoint groups. It is also true for many
“intermediate” groups, like the orthogonal groups in even dimension
which is neither simply connected nor adjoint.
33
The following theorem had been conjectured by A. Weil.
Theorem 4 (Kottwitz, 1988) The Tamagawa number of any
semisimple simply connected algebraic group is equal to one:
τ (G) = 1.
for most of the classical groups: case-by-case verification by Weil in
the around 1960; see Adeles and algebraic groups.
for split groups: Langlands 1966, using “his” Eisenstein series for
adele groups.
for quasi-split groups: Lai, 1980
general case: Kottwitz, proving a certain invariance property under
inner twists.
35
2.3
Siegel’s theorem and Tamagawa numbers
The Tamagawa measure on the adelic points of a semisimple
algebraic group G defined over a number field is a certain, canonically
normalized product measure. It induces an invariant measure on the
coset space G(Ak )/G(k), whose volume is actually finite. The
Tamagawa number of G is defined as
τ (G) := | disc K|− dim G/2 vol G(Ak )/G(k).
34
Note on history. Tamagawa introduced the Tamagawa measure and
thus the Tamagawa number of an algebraic group over a number field
in the late 50s. He himself did not publish much about it, but
apparently he knew that Siegel’s theorem is equivalent to τ (SO) = 2.
Tamagawa numbers of algebraic groups were further investigated by
Ono (see the Boulder proceedings of 1965).
Well known are the lecture notes (Princeton 1961) by Weil, where he
calculated τ (G) for the classical groups.
However, to my best knowledge Kneser was the first who had realized
that one can conveniently use the adelic orthogonal group for a proof
of the Minkowski-Siegel formula. This remark is contained as a
footnote on p. 326 already in his 1956 paper.
36
2.4
Further contributions by M. Kneser
To finish this partial overview of Kneser’s work, I want to mention
three doctoral dissertations which were supervised by him. This is my
personal choice.
H.-V. Niemeier, Definite quadratische Formen der Diskriminante 1
und Dimension 24, J. Number Theory 5 (1973), 142–178.
Jürgen Biermann: Gitter mit kleiner Automorphismengruppe in
Geschlechtern von Z-Gittern mit positiv-definiter quadratischer Form,
Dissertation Göttingen 1981
Yuriko Suwa-Bier Positiv definite quadratische Formen mit gleichen
Darstellungsanzahlen, Dissertation Göttingen 1984
In contrast to a first impression, this is not a question about theta
series. It requires a carefully chosen (and eventually computer-based)
decomposition of the 12-dimensional cone of pairs of reduced positive
definite 3 × 3-matrices.
The solution of this problem was eventually given in the following
dissertation, supervised by F. Grunewald:
Alexander Schiemann: Ternäre positiv definite quadratische Formen
mit gleichen Darstellungsanzahlen, Dissertation Bonn 1993
37
38
3
Finally, I want to mention three papers by Kneser himself, which deal
with quadratic forms, but outside the main scope of his works.
Zur Theorie der Kristallgitter,
Math. Annalen 127, 105–106 (1954)
Two remarks on extreme forms, Canadian Journal of Math.
7, 145–149 (1955)
A particular problem: finite quotients of
Bruhat-Tits buildings
In the 1980s: Finite group theorists and geometers work on the classification of
certain classes of (locally) finite incidence geometries belonging to a Coxeter
diagram (and more general diagrams), together with a flag transitive
automorphism group. The maximal flags are called “chambers”, the term
“chamber system” is also common. These geometries locally look like finite
buildings.
Lineare Relationen zwischen Darstellungsanzahlen quadratischer Formen, Math. Annalen 168, 31–39 (1967)
There is an appropriate covering theory for chamber systems (related to group
amalgamations), and the universal 2-cover under rather general assumptions is a
building. If the diagram belongs to the known list of affine
Coxeter-Dynkin-diagrams and the rank is ≥ 4, then this building is known: it is a
Bruhat-Tits building.
39
40
A general reference is the following:
k
a totally real algebraic number field
o, p, kp , op
as before
G ⊂ GL(V )
simply connected semisimple, almost simple over k
William M. Kantor: Finite geometries via algebraic affine buildings,
pp. 37-44 in: Finite Geometries, Buildings and Related Topics (Eds.
W. M. Kantor et al.), Oxford University Press, Oxford 1990
p
a fixed finite place of k s.t. rkp G ≥ 2
k̄p := o/po
the residue field at p
Other contributors: Timmesfeld, Stroth, Ronan, Meixner.
∆ := ∆(G(kp ))
the Bruhat-Tits building of G(kp ).
We maintain the general notation introduced previously; in particular,
we consider the following:
L
a lattice in V s.t. op L =: Lp defines a vertex of ∆
41
The following proposition makes precise the relation between class
numbers and chamber transitivity of discrete groups as indicated on
p. 40 of Kantor’s above-mentioned paper.
Proposition Under the above assumptions, the following properties
of the lattice L (resp. the arithmetic groups Γ, Γ0 ) are equivalent:
• Γ acts chamber transitively on ∆.
• (i) Γ0 acts chamber transitively on ∆0 ,
(ii) hG (L) = 1.
anisotropic at the infinite places
∆0 ∼
= ∆(G(k̄p )) the residue (star, link) of L in ∆.
Γ := G(o[ 1p ])
a {p}-arithmetic discrete subgroup of G(kp )
Γ0 := G(o)
the finite stabilizer of L in G(k).
42
Proof: (M. Kneser) “=⇒”: (i) is obvious from the assumption,
since the chambers of ∆0 are exactly the chambers of ∆ containing
the vertex L. For (ii), we have to show that
G(Ak ) = G(k) · G(A(∞)).
Since G is isotropic at p, we can use strong approximation for the set
of places ∞ ∪ {p}:
G(Ak ) = G(k) · G(A(∞ ∪ {p})).
(2)
Since Γ acts chamber transitively on ∆ it also acts vertex transitively
on the vertices of a given type, which for “type L” translates as
1
G(kp ) = Γ · G(op ) = G(o[ ]) · G(op ).
p
43
(1)
44
(3)
Given an arbitrary adele (σℓ ) ∈ G(Ak ), first use (2) and write it as
σ · (τℓ ) with σ ∈ G(k) and τℓ ∈ G(oℓ ) for all ℓ 6= p. Then use (3) and
write τp = γ · δ with γ ∈ G(o[ 1p ]) and δ ∈ G(op ). Now replace the
original decomposition of (σℓ ) by
σℓ = (σγ) · (γ −1 τℓ ) for all ℓ.
Since p is a unit in all oℓ , ℓ 6= p, we have γ ∈ G(oℓ ) for all ℓ 6= p and
thus γ −1 τℓ is still in G(oℓ ). Furthermore, γ −1 τp = δ is in G(op ) by
construction. Thus the second factor of the new decomposition is in
G(oℓ ) for all ℓ, and therefore the given adele is a member of the right
hand side of (1).
45
Notice: This proof is completely analogous to the derivation of the
neighbour method from strong approximation. See in particular
equation (3) and compare Kneser’s 1957 paper.
Consequences: Discrete chamber transitive groups on affine
buildings are very rare. Examples had been found in the works of
Kantor and Meixner/Wester in the above-mentioned context.
“⇐=”: Because of assumption (i), we only have to show the
transitivity of Γ on the vertices of “type L”, that is, the vertices in
the orbit G(kp )L ⊂ ∆. But this transitivity is equivalent to (3), as
has already been used. To prove (3), just apply assumption (1) to
adeles which are 1 outside p: for any given σp ∈ G(kp ), there exists
σ ∈ G(k) and an adele (τℓ ) with τℓ ∈ G(oℓ ) for all ℓ s.t. σp = σ · τp
and σ · τℓ = 1 for all ℓ 6= p. But this means σ ∈ G(op ) for all ℓ 6= p,
thus σ ∈ G(o[ 1p ]), as desired.
46
above). It uses only condition (i) (or rather the chamber transtivity
on residues of all types). This condition is already very restrictive, by
a theorem of Gary Seitz. A complete proof of the classification is not
published.
The “generic case” of the suggested proof deals with the
non-existence of such a subgroup for almost all algebraic groups G. It
is briefly sketched in that announcement (see also the survey quoted
A proof of the finiteness result based on the computation of covolumes of
S-arithmetic groups has been given by Prasad and Borel/Prasad (two
subsequent papers in Publ. Math. IHES) already in 1989. It is an open
question whether one could use the class number condition effectively for a
revision of the KLT-classification. For spin groups, one could use the
results on the growth of class numbers of the 1972 dissertation of Ulf
Rehmann, partially based on the 1969 dissertation of Horst Pfeuffer (both
students of Kneser). For other groups, it would apparently be necessary to
control effectively the numerous parameters in the above-mentioned
formulas of (Borel and) Prasad on covolumes of arithmetic groups.
47
48
A full classification has been announced in
[KLT] W.M. Kantor, R. Liebler, J. Tits, On discrete chamber
-transitive automorphism groups of affine buildings,
Notices of the AMS Vol. 16, No. 1 (1987).